With the invention of many new applications such as social network and cloud storage, attribute-based encryption has been studied and applied widely because of its great flexibility, high efficiency and high security. As the current existing attributed-based encryption schemes are most selectively secure, which can’t meet the need of the reality well, how to construct fully secure ABE has been the focus of cryptography. Aimed at the problems mentioned above, an key-policy ABE scheme is firstly constructed by using the dual encryption system in this paper, then the scheme is proved to fully secure in the standard model with the new ideas proposed by Lewko and Waters. Finally the comparison results show that the public and private key lengths of our scheme are similar to the selectively secure GPSW scheme, but our scheme is more secure. Compared with the Lewko-Waters scheme, our scheme has the same security, but has shorter public and private key lengths, which is more efficient. What’s more, similar to the ciphertext-policy scheme of Lekwo-Waters, the techniques of selective security are also utilized in the security proof of our key-policy ABE, which is important in the research of the relation between the selective and full security models.